A COVARIANT CANONICAL DESCRIPTION OF LIOUVILLE FIELD-THEORY

We present a new parameterisation of the space of solutions of Liouvil le field theory on a cylinder. In this parameterisation, the solutions are well-defined and manifestly real functions over all space-time an d all of parameter space. We show that the resulting covariant phase s pace of the Liouville theory is diffeomorphic to the Hamiltonian one, and to the space of initial data of the theory. The Poisson brackets a re derived in our approach, and shown to be those of the co-tangent bu ndle of the loop group of the real line. Using Hamiltonian reduction, we show that our covariant phase space formulation of Liouville theory may also be obtained from the covariant phase space formulation of th e Wess-Zumino-Witten model that we have given previously.